Problema 150 📖Análisis matemático II, Integral indefinida y definida II, Selectividad Mat IIeMatecs Calcula . (Sugerencia: integración por partes). Solución: Como indica la sugerencia utilizaremos la integración por partes: Por tanto: ♦
![\displaystyle\int_0^{\frac \pi 4}\frac x{\cos^2x}~dx=\left.x\,\mbox{tg}\,x\right|_0^{\frac \pi 4}-\int_0^{\frac \pi 4}\mbox{tg}\,x~dx=\\\\=\left.x\,\mbox{tg}\,x\right|_0^{\frac \pi 4}+\int_0^{\frac \pi 4}\frac{-\mbox{sen}\,x}{\cos x}~dx=\\\\=\left.x\,\mbox{tg}\,x\right|_0^{\frac \pi 4}+\ln|\cos x|_0^{\frac \pi 4}=\\\\=\left[x\,\mbox{tg}\,x+\ln|\cos x|\right]_0^{\frac \pi 4}=\\\\=\left(\frac \pi 4\cdot 1+\ln|\sqrt 2/2|\right)-(0)=\\\\=\frac \pi 4+\ln\left(\frac{\sqrt 2}2\right)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cint_0%5E%7B%5Cfrac+%5Cpi+4%7D%5Cfrac+x%7B%5Ccos%5E2x%7D%7Edx%3D%5Cleft.x%5C%2C%5Cmbox%7Btg%7D%5C%2Cx%5Cright%7C_0%5E%7B%5Cfrac+%5Cpi+4%7D-%5Cint_0%5E%7B%5Cfrac+%5Cpi+4%7D%5Cmbox%7Btg%7D%5C%2Cx%7Edx%3D%5C%5C%5C%5C%3D%5Cleft.x%5C%2C%5Cmbox%7Btg%7D%5C%2Cx%5Cright%7C_0%5E%7B%5Cfrac+%5Cpi+4%7D%2B%5Cint_0%5E%7B%5Cfrac+%5Cpi+4%7D%5Cfrac%7B-%5Cmbox%7Bsen%7D%5C%2Cx%7D%7B%5Ccos+x%7D%7Edx%3D%5C%5C%5C%5C%3D%5Cleft.x%5C%2C%5Cmbox%7Btg%7D%5C%2Cx%5Cright%7C_0%5E%7B%5Cfrac+%5Cpi+4%7D%2B%5Cln%7C%5Ccos+x%7C_0%5E%7B%5Cfrac+%5Cpi+4%7D%3D%5C%5C%5C%5C%3D%5Cleft%5Bx%5C%2C%5Cmbox%7Btg%7D%5C%2Cx%2B%5Cln%7C%5Ccos+x%7C%5Cright%5D_0%5E%7B%5Cfrac+%5Cpi+4%7D%3D%5C%5C%5C%5C%3D%5Cleft%28%5Cfrac+%5Cpi+4%5Ccdot+1%2B%5Cln%7C%5Csqrt+2%2F2%7C%5Cright%29-%280%29%3D%5C%5C%5C%5C%3D%5Cfrac+%5Cpi+4%2B%5Cln%5Cleft%28%5Cfrac%7B%5Csqrt+2%7D2%5Cright%29&bg=ffffff&fg=141412&s=0&c=20201002)